**6. Results and Discussion**

Numeric simulations are executed to examine the mixed convective stream and energy transfer of nanoliquids in a wall-driven enclosed box with thermal radiation and entropy generation. The average and cup mixing temperature and its RMSD values are also calculated. The calculations are carried out for a Richardson number (*Ri*) ranging from 0.01 to 102, a volume fraction (φ) of nanoparticles from 0–4 and a radiation parameter from 0 to 10. The Grashof number is used as 10<sup>4</sup> and the Reynolds number varies from 10 to 103. The Prandtl number is taken as Pr = 6.7. The influence of convective stream and energy transport are assessed for several values of the volume fraction of nanoparticles, Richardson number, radiation parameter and the moving-wall directions. The results are depicted graphically for various combinations of parameters and the discussions are given below.

Figure 2 depicts the stream arrangemen<sup>t</sup> for several values of the pertinent parameters Rd and Ri for Case 1 (*U*0 = +1) with Φ = 0.02. In Case 1, wall is moving towards the right side, whereas the lid moves from the right-side to left-side in Case 2. The moving-wall direction is very important and produces the shear force with the adjoining fluid along the upper portion of the box. Since the convective flow is driven by both the buoyant force and the shear stress due to the moving lid, the Richardson number clearly demonstrates the three regimes of convection (free, mixed and forced). The single clockwise rotating eddy appears in the forced convective regime (Ri < 1) for all given values of the radiation parameter. Due to the strong shear force, the core area of the eddy travels towards the right–top corner of the enclosed box. When Ri = 1, that is, in the combined convective regime, the magnitude of both forces (shear and buoyancy) are comparable, the core region moving the center part of the portion of the enclosed box. In the buoyant convective regime, that is, Ri = 100, the variation on the flow pattern is clearly visible here. There is no change on the stream pattern in the forced convection regime when changing the radiation parameter. However, the evidence on the effect of the radiation parameter is clearly seen in the buoyancy convection regime upon raising the values of the radiation parameter for Case 1.

Figure 3 exhibits the convective stream for several values of Ri and Rd for Case 2, with Φ = 0.02. The flow pattern is completely different from Case 1. The dual cell gets for all values of Ri and Rd as it occupies the entire box. Since the shear force is dominant at Ri = 0.01, the core section of the eddy moves towards the left–top corner of the enclosed box. The counter acting eddy could not occupy the whole space as in Case 1, where the movement of liquid particles is aiding with the buoyant force. The buoyant force by the hot liquid along the hot wall produces the clockwise-rotating eddy along the hot wall. However, the shear force dominates here, the eddy by the moving-lid occupies most the box. When rising the Richardson number values to Ri = 1, the mixed convection exists, where both the shear and buoyancy forces are comparable, and the eddy produced by these forces occupies about half of the enclosure in the situation. The natural convection mode at Ri = 100 depicts different phenomena on the stream pattern compared to the other two modes. The eddy by the buoyancy force dominates and occupies most the enclosed box. It is also detected that the eddy by the shear force is weakened on raising the values of the radiation parameter.

**Figure 2.** Streamlines for different Rd and Ri values with U0 = +1 (Case 1), and Φ = 0.02. (**a**) Rd = 0; (**b**) Rd = 1; (**c**) Rd = 5; (**d**) Rd = 10.

**Figure 3.** Streamlines for different Rd and Ri values with U0 = −1 (Case 2) and Φ = 0.02. (**a**) Rd = 0; (**b**) Rd = 1; (**c**) Rd = 5; (**d**) Rd = 10.

Figure 4 depicts the thermal distribution for several values of the radiation parameter and the Richardson number for Case 1 with Φ = 0.02. The thermal boundary layers are shaped along the hot wall for all assumed values of the radiation parameter in the forced convection regime. The temperature boundary layers is weakened for higher values of the radiation (Rd = 10) in the combined convection regime. The horizontal thermal stratification appears in the central region of the enclosed box in the absence of radiation or lower values of the radiation parameter for the natural convection regime. The temperature gradients near wall(s) disappear on rising the value of the radiation parameter. Figure 5 exhibits the isotherms for an opposite moving lid (Case 2) with the same parameters in Figure 4. The thermal layers at the boundary do not appear along the hot wall in forced convection regime as in Case 1. Due to the dual cell structure in the flow field, the thermal layers at the boundary are collapsed along the hot wall in Case 2.

Figure 6 depicts the drag coe fficient for several values of Rd and Ri for both cases of the moving lid directions. In Case 1, the skin friction declines upon raising the values of Ri. However, in Case 2, the skin friction behaves nonlinearly, that is, the skin friction grows up to Ri = 1 and then it declines upon raising the values of Ri. It is detected that there is no change on the averaged skin friction for numerous values of Rd when Ri = 0.01 and Ri = 0.1, that is, in the forced convective regime. The skin friction declines upon rising the values of the radiation parameter in the combined and natural convective regimes.

Since the energy transport rate is a key factor in the thermal systems, the (average) energy transfer rate is depicted via the Nusselt number to explore the e ffect on various pertinent parameters. The local energy transport along the heated wall is computed by the local Nusselt number and it is depicted in Figure 7 for both cases of moving-wall directions. It is clearly exhibited from Figure 7a,c,e that the energy transport is diminished upon raising the values of the Richardson number for Case 1. That is, the local energy transport along the hot wall is enhanced in the forced convective regime. It is almost thrice the value of local Nusselt number for free-convection regime. Case 2 also provides a similar trend on the energy transport upon raising the values of Ri number. It is detected that the local heat transport rises upon raising the radiation parameter for all convection regimes. The highest local energy transfer is observed at the bottom of the heated wall for Case 1 and then it decreases along the wall height. However, the highest local energy transfer is detected at the top of the heated wall for Ri = 0.01 and Ri = 1 in Case 2. However, the opposite trend is found for the free-convection regime in Case 2. The moving lid direction supports the fluid motion with the aiding of the buoyancy force. However, in Case 2, the moving lid direction suppresses the buoyancy force at the top section of the heated wall, and it results the dual cellular motion inside the enclosure. In the dual cell structure, the two cells hit at the top–left corner and provides the highest local heat energy transfer at this point, which is clearly seen from Figure 7b. The high amount of shear force has driven the heated fluid particles vigorously at this situation. Hence, the local heat energy transfer gives a similar trend in both cases for the natural convection regime.

**Figure 4.** Isotherms for diverse Rd and Ri values with U0 = +1 (Case 1) and Φ = 0.02. (**a**) Rd = 0; (**b**) Rd = 1; (**c**) Rd = 5; (**d**) Rd = 10.

**Figure 5.** Isotherms for different Rd and Ri values with U0 = −1 (Case 2) and Φ = 0.02. (**a**) Rd = 0; (**b**) Rd = 1; (**c**) Rd = 5; (**d**) Rd = 10.

**Figure 6.** Drag coefficient versus Ri for different Rd. (**a**) Case 1; (**b**) Case 2.

**Figure 7.** Local Nusselt number for diverse Ri and Rd. (**<sup>a</sup>**,**c**,**<sup>e</sup>**) Case 1; (**b**,**d**,**f**) Case 2.

Figure 8 demonstrates the averaged Nusselt number for several values of Rd and Ri for Case 1 (*U*0 = +1) and Case 2 (*U*0 = −1). The averaged heat transport rate is enhanced upon raising the values of the radiation parameter for both cases of the moving-wall directions. It is detected that the averaged heat transfer declines upon raising the values of Ri. Further, scrutinizing these figures, it is found that the moving-wall direction affects the thermal energy transfer rate evidently. When the wall moves from the right-side to left-side (Case 2), the heat energy transfer rate is less due to the dual-eddy structure. The effect of nanometer sized particle volume fraction on the averaged energy transport is examined and it is portrayed in Figure 9a,b for several values of the Richardson number and two cases of moving-wall directions in the presence of radiation with Rd = 5. The averaged heat transport rate decreases upon raising the values of the nanoparticle volume fraction from 0%~4% in mixed and free convective regimes for both moving-wall cases. But, the averaged heat transport rate rises with the nanoparticles volume fraction in Case 1 at Ri = 0.01. In Case 2 at Ri = 0.01, the averaged heat transfer increases first up to Φ = 2% and then it decreases upon raising the value of Φ. Comparing these two cases in Figure 9a,b, it is detected that the averaged Nusselt number is always high for Case 1 than that of Case 2. This is because of the dual eddy structure in Case 2. The energy transfer from the hotter region to the colder region taken by a single cell is faster than the energy transport by the two cells inside the enclosed box. Since the energy exchange between the two cells takes some time which slows down the overall energy transport within the enclosed box.

**Figure 8.** Averaged Nusselt number versus Ri for different Rd. (**a**) Case 1; (**b**) Case 2.

**Figure 9.** Averaged Nusselt number versus Φ for different Ri with Rd = 5. (**a**) Case 1; (**b**) Case 2.

Figure 10 shows the increment level of the averaged energy transport for different radiation values compared with the absence of radiation parameter. The data clearly show the increasing level of averaged energy transport while raising the values of Rd in both cases of moving wall. The increment level is very high in the natural convection regime in both cases. Figure 11 demonstrates the cup-mixing temperature for various values of Ri and Rd parameters. The behavior of cup-mixing temperature is nonlinear fashion for Case 1, however, Case 2 shows almost a linear fashion. The deviation in cup-mixing temperature with Rd is high at forced convection regime for Case 2. However, it is almost same in free-convection case. The Tcup values are almost constant when changing the values of Rd in free-convection flow for Case 2. Figure 12 demonstrates the average temperature for different Ri and Rd values. The higher Tcup values indicates the well mixing of fluid with higher temperature. It is obviously seen from Figure 12 that the Tavg is almost constant for all values of Rd in free-convection regime. The maxima of Tavg attains at Ri = 100 for all Rd values in Case 1, see Figure 12a. From Figure 12b, we observe that the deviation of Tavg is high at Ri = 0.01 in Case 2.

**Figure 10.** Increment of averaged Nusselt number. (**a**) Case 1; (**b**) Case 2.

**Figure 11.** Cup-mixing temperature for different Ri and Rd values. (**a**) Case 1; (**b**) Case 2.

Figures 13 and 14 portray the RMSDTcup and RMSDTavg for both cases with different Ri and Rd values. Since the nondimensional temperature varies between 0 and 1, the RMSD values are below 1 in the present examination. It is noticed from Figure 13a that the RMSDTcup increases first and decreases on raising the Ri values for Case 1. The opposite trend is observed for Case 2 in the absence of thermal radiation. However, RMSDTcup increases linearly with Ri for Rd ≥ 5 for Case 2. It is observed from Figure 14, RMSDTavg rises linearly with Richardson number in Case 1 for all values of Rd. It is also detected from Figure 14 that the RMSDTavg rises when growing the Rd values. However, in Case 2, it behaves nonlinearly for either absence of Rd or low values of Rd. However, it acts as same as Case 1 for higher values of Rd (≥5). RMSDTavg attains its maxima at strong free-convection region in the presence of thermal radiation. Since the RMSD values are lower in all cases, we ge<sup>t</sup> higher temperature uniformity inside the box.

**Figure 12.** Averaged temperature for different Ri and Rd values. (**a**) Case 1; (**b**) Case 2.

**Figure 13.** RMSDTcup for different Ri and Rd values. (**a**) Case 1; (**b**) Case 2.

**Figure 14.** RMSDTavg for different Ri and Rd values. (**a**) Case 1; (**b**) Case 2.

Figure 15 portrays the influence of Bejan number for both cases with different Ri and Rd values. The values of Be are almost constant on raising the Ri values until Ri = 10, but, after this, it suddenly fall down at Ri = 100 for both direction of moving-wall. When raising the Rd values, the Bejan number is increased. It results that the radiation parameter boosted up the entropy generation inside the box. It is clear that Be lies between 0 and 1. If Be tends to 0 then the irreversibility due to fluid friction controls. If Be tends to 1, the irreversibility due to thermal transfer is leading. In all cases, the values of Be is tends to 1, it results that the irreversibility due to thermal transfer is dominant here.

**Figure 15.** Bejan number for different Ri and Rd values. (**a**) Case 1; (**b**) Case 2.
